$(X_{1},Y_{1}),\,...\,,(X_{n},Y_{n})' (n>2)$ are random samples taken from
$N_{2}((\mu_{1},\,\mu_{2})',\,$$ \begin{pmatrix} \sigma^{2}_{1} & \rho\sigma_{1}\sigma_{2} \\ \rho\sigma_{1}\sigma_{2} & \sigma^{2}_{2} \\ \end{pmatrix}) $$ $
$r$ is sample correlation coefficient.
Also, $Z_{i}=(X_{i}-\mu_{1})/\sigma_{1},\,W_{i}=[(Y_{i}-\mu_{2})/\sigma_{2}-\rho(X_{i}-\mu_{1})]/\sqrt{1-\rho^{2}}\\ S_{zz}=\sum_1^n(Z_{i}-\overline Z)^{2},\,S_{ww}=\sum_1^n(W_{i}-\overline W)^{2},\,S_{zw}=\sum_1^n(Z_{i}-\overline Z)(W_{i}-\overline W)$
I proved Z & W are independent and $S_{ww}-S^{2}_{zw}/S_{zz}\,=\,W'(I-1(1'1)^{-1}1'-T(T'T)^{-1}T')W,\,\, where\,\, T=(Z-1\overline Z)/\sqrt {S_{zz}}$
I want to show $1)\, S_{ww}-S^{2}_{zw}/S_{zz}$ ~ $\chi^{2}(n-2),$ and independent from Z.
$2)\, S_{zw}/\sqrt {S_{zz}}$ ~ $N(0, 1)$ and independent from Z.
If i consider $\,\,T(T'T)^{-1}T'$ as constant, then it is easy to derive the distribution of $S_{ww}-S^{2}_{zw}/S_{zz}$. However, $\,\,T(T'T)^{-1}T'$ is a function of Z, so that is a random variable, which makes it difficult for me to derive the distribution. Would you help me?